The real numbers: Pythagoras to Stevin

Version for printingBefore we begin to discuss the historical development of the real number system it is useful to consider what a number is. Perhaps the reader might think that this is a silly question and that it is "obvious" what a number is. Well the first clear evidence that this is not so is the fact that the concept of number has changed greatly throughout the development of mathematics up to the present day. What is equally clear is that there is no reason, other than conceit, to believe that the present concept of number will not change in the future. Wittgenstein, in Philosophical Investigations writes:-

Why do we call something a 'number'? Well, perhaps because it has a direct relationship with several things that have hitherto been called number; and this can be said to give it an indirect relationship to other things we call the same name. And we extend our concept of number as in spinning a thread we twist fibre on fibre. And the strength of the thread does not reside in the fact that some one fibre runs through its whole length, but in the overlapping of many fibres. But if someone wished to say: "There is something common to all these constructions - namely the disjunction of all their common properties" - I should reply: "Now you are only playing with words. One might as well say: 'Something runs through the whole thread - namely the continuous overlapping of those fibres.' "

"All right: the concept of number is defined for you as the logical sum of these individual interrelated concepts: cardinal numbers, rational numbers, real numbers etc.; and, in the same way the concept of a game is the logical sum of a corresponding set of sub-concepts." - It need not be so. For I can give the concept 'number' rigid limits in this way, that is, use the word "number" for a rigidly limited concept, but I can also use it so that the extension of the concept is not closed by a frontier. And this is how we use the word "game". For how is the concept of a game bounded?

We should begin a discussion of real numbers by looking at the concepts of magnitude and number in ancient Greek times. The first of these might refer to the length of a geometrical line while the second concept, namely number, was thought of as composed of units. Pythagoras seems to have thought that "All is number"; so what was a number to Pythagoras? It seems clear that Pythagoras would have thought of 1, 2, 3, 4, ... (the natural numbers in the terminology of today) in a geometrical way, not as lengths of a line as we do, but rather in the form of discrete points. Addition, subtraction and multiplication of integers are natural concepts with this type of representation but there seems to have been no notion of division. A mathematician of this period, given the number 12, could easily see that 4 is a submultiple of it since 3 times 4 is exactly 12. Although to us this is clearly the same as division, it is important to see the distinction. We have used the word "submultiple" above, so should indicate what the Pythagoreans considered this to be. Nicomachus, following the tradition of Pythagoras, makes the following definition of a submultiple:-

The submultiple, which is by its nature the smaller, is the number which when compared with the greater can measure it more times than one so as to fill it out exactly.

Magnitudes, being distinct entities from numbers, had to have a separate definition and indeed Nicomachus makes such a parallel definition for magnitudes.

[In the time of Pythagoras] since all other things seemed in their whole nature to be modelled on numbers, and numbers seemed to be the first things in the whole of nature, they supposed the elements of numbers to be the elements of all things, and the whole heaven to be a musical scale and a number. And all the properties of numbers and scales which they could show to agree with the attributes and parts and the whole arrangement of the heavens, they fitted into their scheme ... the Pythagoreans say that things are what they are by intimating numbers ... the Pythagoreans take objects themselves to be numbers and do not treat mathematical objects as distinct from them ...

This concept certainly ran into difficulties once various magnitudes were studied. All numbers, essentially by definition, were, as we have seen, (positive integer) multiples of a base unit but ratios of lengths were shown not to have the property of being ratios of numbers (integers). The usual example given of this comes from a right angled triangle whose shorter sides are both of unit length. Such a triangle has as hypotenuse a line of length √2 times the lengths of the shorter sides. There is no length x such that 1 and √2 are both multiples (remember integer multiples) of x. Plato, in Theaetetus, tells of the discovery that √3, √5, ... , √17 were not commensurable with 1:-

Theodorus was writing out for us something about roots, such as the sides of squares whose area was 3 or 5 units, showing that the sides are incommensurable with the unit: he took the examples up to 17, but there for some reason he stopped.

We suppose that the discovery that √2 was not commensurable with 1 came earlier which is why Theodorus started with √3. Heimonen, in [10], looks at the views of different historians concerning the discovery of the irrational numbers:-

Von Fritz has proposed that the Pythagorean Hippasos first proved the irrationality of the golden ratio by studying the regular pentagon. The proof is based on the fact that the continued fraction expansion of the ratio of its diagonal and size is periodic.

The same idea of irrationality proof was expressed by Zeuthen and van der Waerden for the ratio of the diagonal and side of the square also, as well as for the square roots of 3, 5, ... ,17, which according to Plato were proved to be irrational by Theodoros.

Knorr set out a new theory, trying especially to explain better why Theodoros stopped just at the square root of 17. His theory is some kind of geometrical version on the irrationality proof of the square root of 2 known from school.

Fowler accepted the main ideas of Knorr, but also returned to the continued fractions, maintaining even that also the common fractions were handled as continued fractions in Plato's time.

Before continuing to describe advances in ideas concerning numbers, it should be mentioned at this stage that the Egyptians and the Babylonians had a different notion of number to that of the ancient Greeks. The Babylonians looked at reciprocals and also at approximations to irrational numbers, such as √2, long before Greek mathematicians considered approximations. The Egyptians also looked at approximating irrational numbers.

Let us now look at the position as it occurs in Euclid's Elements. This is an important stage since it would remain the state of play for nearly the next 2000 years. In Book V Euclid considers magnitudes and the theory of proportion of magnitudes. It is probable (and claimed in a later version of The Elements) that this was the work of Eudoxus. Usually when Euclid wants to illustrate a theorem about magnitudes he gives a diagram representing the magnitude by a line segment. However magnitude is an abstract concept to Euclid and applies to lines, surfaces and solids. Also, more generally, Euclid also knows that his theory applies to time and angles.

Given that Euclid is famous for an axiomatic approach to mathematics, one might expect him to begin with a definition of magnitude and state some unproved axioms. However he leaves the concept of magnitude undefined and his first two definitions refer to the part of a magnitude and a multiple of a magnitude:

Definition V.1 A magnitude is a part of a magnitude, the less of the greater, when it measures the greater.

Again the term "measures" here is undefined but clearly Euclid means that (in modern symbols) the smaller magnitude x is a part of the greater magnitude y if nx = y for some natural number n > 1.

Definition V.2 The greater is a multiple of the less when it is measured by the less.

Then come the definition of ratio.

Definition V.3 A ratio is a sort of relation in respect of size between two magnitudes of the same kind.

This is an exceptionally vague definition of ratio which basically fails to define it at all. He then defines when magnitudes have a ratio, which according to the definition is when there is an multiple (by a natural number) of the first which exceeds the second and a multiple of the second which exceeds the first. Then comes the vital definition of when two magnitudes are in the same ratio as a second pair of magnitudes. As it is quite hard to understand in Euclid's language, let us translate it into modern notation. It says that a : b = c : d if given any natural numbers n and m we have

na > mb if and only ifnc > mdna = mb if and only if nc = mdna < mb if and only if nc < md.

Euclid then goes on to prove theorems which look to a modern mathematician as if magnitudes are vectors, integers are scalars, and he is proving the vector space axioms. For example for magnitudes a and b and natural numbers n and m he proves:-

In Book VII Euclid studies numbers. He makes a series of definitions. First he defines a unit, then a number is defined as being composed of a multitude of units, and parts and multiples are defined as for magnitudes. We should note that Euclid, as earlier Greek mathematicians, did not consider 1 as a number. It was a unit and the numbers 2, 3, 4, ... were composed of units. Various properties of numbers are assumed but are not listed as axioms. For example the commutative law for multiplication is assumed without ever being stated as an axiom as are the associative law for addition etc. He then introduces proportion for numbers and shows essentially that for numbers a, b, c, d that a : b = c : d precisely when the least numbers with ratio a : b is equal to the least numbers with ratio c : d. This is logically equivalent to saying in modern terms that the rational a/b and the rational c/d are equal if they become the same when reduced to their lowest terms. An important result in Book VII is the Euclidean algorithm. We should note that Euclid never identified the ratio 2 : 1 with the number 2. These were two quite different concepts.

Book X considers commensurable and incommensurable magnitudes. It is a long book, over one quarter of the whole of The Elements. We have:

Definition X.1 Those magnitudes are said to be commensurable which are measured by the same measure, and those incommensurable which cannot have any common measure.

Proposition X.2 If, when two unequal magnitudes are set out and the lesser is always subtracted in turn from the greater, the remainder never measures the magnitude before it, then the magnitudes will be incommensurable.

Proposition X.5 Commensurable magnitudes have to one another the ratio which a number has to a number.

Notice that Proposition X.2 says that two magnitudes are incommensurable if the Euclidean algorithm does not terminate. Euclid goes on to prove, among many other results, those of Theodorus, namely that segments of length √3, √5, ... , √17 are incommensurable with a segment of unit length.

So where does Euclid's Elements leave us with respect to numbers. Basically numbers were 1, 2, 3, ... and ratios of numbers were used which (although not considered to be numbers) basically allowed manipulation with what we call rationals. Also magnitudes were considered and these were essentially lengths constructible by ruler and compass from a line of unit length. No other magnitudes were considered. Hence mathematicians studied magnitudes which had lengths which, in modern terms, could be formed from positive integers by addition, subtraction, multiplication, division and taking square roots.

The Arabic mathematicians went further with constructible magnitudes for they used geometric methods to solve cubic equations which meant that they could construct magnitudes whose ratio to a unit length involved cube roots. For example Omar Khayyam showed how to solve all cubic equations by geometric methods. Fibonacci, using skills learnt from the Arabs, solved a cubic equation showing that its root was not formed from rationals and square roots of rationals as Euclid's magnitudes were. He then went on to compute an approximate solution. Although no conceptual advances were taking place, by the end of the fifteenth century mathematicians were considering expressions build from positive integers by addition, subtraction, multiplication, division and taking nth roots. These are called radical expressions.

By the sixteenth century rational numbers and roots of numbers were becoming accepted as numbers although there was still a sharp distinction between these different types of numbers. Stifel, in his Arithmetica Integra (1544) argues that irrationals must be considered valid:-

It is rightly disputed whether irrational numbers are true numbers or false. Because in studying geometrical figures, where rational numbers desert us, irrationals take their place, and show precisely what rational numbers are unable to show ... we are moved and compelled to admit that they are correct ...

However, he goes on to argue that, as they are not proportional to rational numbers, they cannot be true numbers even if they are correct. He ends up arguing that all irrational numbers result from radical expressions. Well the obvious question the reader might feel they want to ask Stifel is: what about the length of the circumference of a circle with radius of unit length? In fact Stifel gives an answer to this in an appendix to the book. First he makes a distinction between physical circles and mathematical circles. One can measure the properties of physical circles, he claims, but one cannot measure a mathematical circle with physical instruments. He then goes on to consider the circle as the limit of a sequence of polygons of more and more sides. He writes:-

Therefore the mathematical circle is rightly described as the polygon of infinitely many sides. And thus the circumference of the mathematical circle receives no number, neither rational nor irrational.

Not too good an argument, but nevertheless a remarkable insight that there were lengths which did not correspond to radical expressions but which could be approximated as closely as one wished.

A major advance was made by Stevin in 1585 in La Theinde when he introduced decimal fractions. One has to understand here that in fact it was in a sense fortuitous that his invention led to a much deeper understanding of numbers for he certainly did not introduce the notation with that in mind. Only finite decimals were allowed, so with his notation only certain rationals to be represented exactly. Other rationals could be represented approximately and Stevin saw the system as a means to calculate with approximate rational values. His notation was to be taken up by Clavius and Napier but others resisted using it since they saw it as a backwards step to adopt a system which could not even represent 1/3 exactly.

Stevin made a number of other important advances in the study of the real numbers. He argued strongly in L'Arithmetique (1585) that all numbers such as square roots, irrational numbers, surds, negative numbers etc should all be treated as numbers and not distinguished as being different in nature. He wrote:-

Thesis 1: That unity is a number.
Thesis 2: That any given numbers can be square, cubes, fourth powers etc.
Thesis 3: That any given root is a number.
Thesis 4: That there are no absurd, irrational, irregular, inexplicable or surd numbers.

It is a very common thing amongst authors of arithmetics to treat numbers like √8 and similar ones, which they call absurd, irrational, irregular, inexplicable or surds etc and which we deny to be the case for number which turns up.

His first thesis was to argue against the Greek idea that 1 is not a number but a unit and the numbers 2, 3, 4, ... were composed of units. The other three theses were encouraging people to treat different types of numbers, which were at that time treated separately, as a single entity - namely a number.

One further comment by Stevin in L'Arithmetique is worth recording. He noted that, as we stated above, Euclid's Proposition X.2 says that two magnitudes are incommensurable if the Euclidean algorithm does not terminate. Stevin writes about this pointing out what today we would say was the difference between an algorithm and a procedure (or semi-algorithm):-

Although this theorem is valid, nevertheless we cannot recognise by such experience the incommensurability of two given magnitudes. ... even though it were possible for us to subtract by due process several hundred thousand times the smaller magnitude from the larger and continue that for several thousands of years, nevertheless if the two given numbers were incommensurable one would labour eternally, always ignorant of what could still happen in the end. This manner of cognition is therefore not legitimate, but rather an impossible position ...

Further progress in the development of the real numbers only became possible after ideas of convergence were put on a firm basis. However, there was a strong influence in the other direction too, since progress in rigorous analysis required a deeper understanding of the real numbers. This is studied further in the article The real numbers: Stevin to Hilbert.